I devise a numerical method of high order in space (FDMHS) to simulate flow past a finite plate and a semi-infinite plate. The method solves the incompressible Navier-Stokes equation in the stream function-vorticity
formulation. The focus is to study a fundamental problem in fluid dynamics, namely, flow past sharp edges. Resolving this flow structure is difficult, in
particular at early times. The difficulty is due to the fact that large velocity gradients and vorticity are present in a very thin boundary layer attached to the plate initially. FDMHS is a splitting method, implicit in
time and uses compact fourth order finite differences. FDMHS has demonstrated satisfactory performance in our numerical simulations.
For the finite plate case, three background flow are used: impulsively started,
uniformly accelerated, and oscillating. Resolved computations show structure of the boundary layer separation and roll-up from very early times to relative large times.
For the impulsively started, the details of vorticity structure at
early times have been studied. We resolved the region of negative vorticity along the plate induced by and entrained into the leading vortex. A secondary entrainment of positive vorticity into the region of negative vorticity is also found. The maximum velocity decays as $t^{-1/4}$ over a large initial time interval.
For the uniformly accelerated, we show evolution in the appropriate
non-dimensional variables, and find agreement with scaling laws observed in experiments.
For the oscillating, we compared the viscous simulation using FDMHS
with an inviscid vortex sheet method. Both are in excellent
agreement at early times. There are difference at later times. most likely caused by wall vorticity which is not accounted for by the vortex sheet model.
The shed circulation is independent of viscosity initially for all three background flows. The effect of viscosity on the vorticity evolution and
on quantities such as the shed circulation, core trajectory and vorticity, vortex size and width are also presented. For the semi-infinite plate case, we derived the scaling rule and verified it numerically.